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Background to Sound waves and Resonance
Note: this page should be written in a style and language suitable for A-Level students. * What are waves (general physical and mathematical properties of sinusoidal waves) * Waves in the context of sound (Pressure and particle distribution) * To enter maths on the page use LaTeX: https://en.wikipedia.org/wiki/Help:Displaying_a_formula What are waves? A wave is a disturbance in both space and time; an oscillation carries that transfers energy. There are two types of wave, transverse and longitudinal. A transverse wave is arguably the easiest to imagine because we see these most commonly, a prime example being a wave on the surface of the sea. This is a transverse wave because the molecules are oscillating perpendicular to the plane of movement of the wave. This can be shown by Figure 1, a wave on the sea, where the waves can be seen moving in time as well as moving water towards the shore. All waves are similar to this, but vary in how they can be described mathematically. For simplicity here, we will only deal with sinusoidal waves. A longitudinal wave is one whose oscillations are parallel to the plane of movement of the way. This occurs as shown in Figures 2 and 3 by compressions and rarefactions. Mathematics of Waves (General basic laws that govern all waves) To describe the motion of a wave physicists use the concept of a wave function which describes the position of a particle in the medium at any time. Since there are many different types of waves, there are many different wave functions. The most basic of these wave functions are sinusoidal waves, which refers to a periodic wave or a wave with repetitive motion. To start developing the wave function we must first define a few inherent properties of the wave, assign them a symbol and give their SI units. These are: * Wave speed (v): The speed of the wave's propagation given the units meters per second. * Amplitude (A): The maximum magnitude of the displacement from equilibrium, given in meters. * Period (T): The time for one full wave cycle to occur in seconds. This can be thought of as the time between two pulses, or from crest to crest / trough to trough. * Frequency (f): The number of cycles in a unit of time, given the unit hertz (Hz) which is equivalent to o one cycle per second. * Angular frequency ( \omega ): This is 2 \pi times the frequency with units of radians per second. * Wavelength ( \lambda ): The distance between any two points at corresponding positions on the successive repetitions in the wave, for example, from one crest or trough to the next with units of meters. *Wave number (k): This is a useful quantity also known as the propagation constant which is defined as 2 \pi divided by the wavelength, giving the units as radians per meter. Now we have the basic quantities defined, we can start building relations between them. From the definitions of period T and frequency f we see that each is the reciprocal of each other giving the relationship: f=\dfrac{1}{T} Similarly, from the definition of angular frequency we can find the relationship: \omega=2\pif=\dfrac{2\pi}{T} Sound waves A sound wave is a vibration that propagates as a mechanical wave of pressure and displacement. Unlike light waves, sound waves cannot travel through a vacuum as they require a medium of matter to propagate through such as air or water. Sound waves can also be longitudinal (Figure 2) or transverse (Figure 5), depending on the medium they are propagating though. For example, if the sound wave propagates through air or water, it will take on a longitudinal form and for solids the wave will be transverse. Generating a sound wave generally involves a vibrating source, such as a stereo speaker. The source causes vibrations in the surrounding medium which, as the source continuously vibrates, propagate away from the source at the speed of sound. If we were to consider a point that is a fixed distance away from the source we would find that the pressure, velocity and displacement of the medium vary in time. Similarly, if we were to freeze time and look at the static wave, the pressure, velocity and displacement of the medium vary in space. One misconception that is often found is that you may think the particles in the medium, for example air, travel with the wave as it propagates. This is not the case as the particles oscillate around their equilibrium point (in other words, their average position) while the wave propagates as if it were superimposed on top of the motion of the particles. This can be seen from figure 2. The particles act as a 'transporter' for the vibrations. While sound waves can be described by the same simple mathematical laws that govern all waves, how the collective motion of all particles in the medium and how the properties of the medium itself correspond to the wave is much more complicated. Firstly, there is a complex relationship between density and pressure of the medium which is affected by temperature. This relationship determines the speed of sound and so, unlike light waves which have constant speed in a vacuum, the speed of sound though a medium is dependent on the properties of the medium. For example, the speed of sound in dry air at 20 degrees Celsius at 'sea level' atmospheric pressure is 343.2 m/s whereas in water it is over 4 times as fast at 1,484 m/s. Secondly, the motion of the medium itself can effect the propagating wave. Any movement may cause an increase or decrease of the speed of the sound wave depending on the direction of the movement. This is best imagined by thinking about a sound wave moving though wind. The speed of propagation will increase if the sound wave moves in the same direction as the wind and will decrease in the sound and wind are moving in opposite directions. Finally, the viscosity of the medium also effects the propagation (Figure 6). For most media, such as air or water, this is a negligible effect, but for medium or greater viscosity a process called attenuation can occur, which is the gradual loss of intensity as the wave propagates, causing the sound to become quieter at a faster rate. Mathematics of Sound Waves (Laws specific to sound waves and equations of how medium properties effect sound wave properties) Sound waves: Advanced (Perhaps include more in-depth information on more complicated properties such as pitch, duration, loudness, timbre, noise ect for 'students' wanting to develop their knowledge beyond what is necessary)